LMFDB - Embedded newform 6036.2.a.g.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.03500$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.0349978 q^{5} +0.637505 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.0349978 q^{5} +0.637505 q^{7} +1.00000 q^{9} -0.863742 q^{11} -4.36921 q^{13} +0.0349978 q^{15} -2.27774 q^{17} -4.02205 q^{19} +0.637505 q^{21} +8.23006 q^{23} -4.99878 q^{25} +1.00000 q^{27} +1.29094 q^{29} -1.21322 q^{31} -0.863742 q^{33} +0.0223113 q^{35} +7.14619 q^{37} -4.36921 q^{39} +2.42495 q^{41} +0.00158582 q^{43} +0.0349978 q^{45} -3.98820 q^{47} -6.59359 q^{49} -2.27774 q^{51} +7.67263 q^{53} -0.0302291 q^{55} -4.02205 q^{57} -1.59240 q^{59} -3.50074 q^{61} +0.637505 q^{63} -0.152913 q^{65} -15.8723 q^{67} +8.23006 q^{69} -12.8739 q^{71} -2.85083 q^{73} -4.99878 q^{75} -0.550640 q^{77} +7.20069 q^{79} +1.00000 q^{81} -13.4282 q^{83} -0.0797159 q^{85} +1.29094 q^{87} -10.0457 q^{89} -2.78539 q^{91} -1.21322 q^{93} -0.140763 q^{95} +11.8338 q^{97} -0.863742 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.0349978	0.0156515	0.00782575	−	0.999969i	$-0.497509\pi$
$5$	0.0349978	0.0156515	0.00782575	+	0.999969i	$0.497509\pi$
$6$	0	0
$7$	0.637505	0.240954	0.120477	−	0.992716i	$-0.461558\pi$
$7$	0.637505	0.240954	0.120477	+	0.992716i	$0.461558\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.863742	−0.260428	−0.130214	−	0.991486i	$-0.541566\pi$
$11$	−0.863742	−0.260428	−0.130214	+	0.991486i	$0.541566\pi$
$12$	0	0
$13$	−4.36921	−1.21180	−0.605900	−	0.795541i	$-0.707187\pi$
$13$	−4.36921	−1.21180	−0.605900	+	0.795541i	$0.707187\pi$
$14$	0	0
$15$	0.0349978	0.00903639
$16$	0	0
$17$	−2.27774	−0.552433	−0.276217	−	0.961095i	$-0.589081\pi$
$17$	−2.27774	−0.552433	−0.276217	+	0.961095i	$0.589081\pi$
$18$	0	0
$19$	−4.02205	−0.922722	−0.461361	−	0.887213i	$-0.652639\pi$
$19$	−4.02205	−0.922722	−0.461361	+	0.887213i	$0.652639\pi$
$20$	0	0
$21$	0.637505	0.139115
$22$	0	0
$23$	8.23006	1.71609	0.858043	−	0.513577i	$-0.171680\pi$
$23$	8.23006	1.71609	0.858043	+	0.513577i	$0.171680\pi$
$24$	0	0
$25$	−4.99878	−0.999755
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.29094	0.239721	0.119860	−	0.992791i	$-0.461755\pi$
$29$	1.29094	0.239721	0.119860	+	0.992791i	$0.461755\pi$
$30$	0	0
$31$	−1.21322	−0.217901	−0.108951	−	0.994047i	$-0.534749\pi$
$31$	−1.21322	−0.217901	−0.108951	+	0.994047i	$0.534749\pi$
$32$	0	0
$33$	−0.863742	−0.150358
$34$	0	0
$35$	0.0223113	0.00377129
$36$	0	0
$37$	7.14619	1.17483	0.587413	−	0.809287i	$-0.300146\pi$
$37$	7.14619	1.17483	0.587413	+	0.809287i	$0.300146\pi$
$38$	0	0
$39$	−4.36921	−0.699633
$40$	0	0
$41$	2.42495	0.378714	0.189357	−	0.981908i	$-0.439360\pi$
$41$	2.42495	0.378714	0.189357	+	0.981908i	$0.439360\pi$
$42$	0	0
$43$	0.00158582	0.000241836 0	0.000120918	−	1.00000i	$-0.499962\pi$
$43$	0.00158582	0.000241836 0	0.000120918		1.00000i	$0.499962\pi$
$44$	0	0
$45$	0.0349978	0.00521716
$46$	0	0
$47$	−3.98820	−0.581739	−0.290870	−	0.956763i	$-0.593945\pi$
$47$	−3.98820	−0.581739	−0.290870	+	0.956763i	$0.593945\pi$
$48$	0	0
$49$	−6.59359	−0.941941
$50$	0	0
$51$	−2.27774	−0.318947
$52$	0	0
$53$	7.67263	1.05392	0.526958	−	0.849891i	$-0.323332\pi$
$53$	7.67263	1.05392	0.526958	+	0.849891i	$0.323332\pi$
$54$	0	0
$55$	−0.0302291	−0.00407609
$56$	0	0
$57$	−4.02205	−0.532734
$58$	0	0
$59$	−1.59240	−0.207313	−0.103657	−	0.994613i	$-0.533054\pi$
$59$	−1.59240	−0.207313	−0.103657	+	0.994613i	$0.533054\pi$
$60$	0	0
$61$	−3.50074	−0.448224	−0.224112	−	0.974563i	$-0.571948\pi$
$61$	−3.50074	−0.448224	−0.224112	+	0.974563i	$0.571948\pi$
$62$	0	0
$63$	0.637505	0.0803181
$64$	0	0
$65$	−0.152913	−0.0189665
$66$	0	0
$67$	−15.8723	−1.93911	−0.969557	−	0.244866i	$-0.921256\pi$
$67$	−15.8723	−1.93911	−0.969557	+	0.244866i	$0.921256\pi$
$68$	0	0
$69$	8.23006	0.990783
$70$	0	0
$71$	−12.8739	−1.52786	−0.763928	−	0.645302i	$-0.776732\pi$
$71$	−12.8739	−1.52786	−0.763928	+	0.645302i	$0.776732\pi$
$72$	0	0
$73$	−2.85083	−0.333665	−0.166832	−	0.985985i	$-0.553354\pi$
$73$	−2.85083	−0.333665	−0.166832	+	0.985985i	$0.553354\pi$
$74$	0	0
$75$	−4.99878	−0.577209
$76$	0	0
$77$	−0.550640	−0.0627512
$78$	0	0
$79$	7.20069	0.810141	0.405070	−	0.914286i	$-0.367247\pi$
$79$	7.20069	0.810141	0.405070	+	0.914286i	$0.367247\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.4282	−1.47394	−0.736970	−	0.675925i	$-0.763744\pi$
$83$	−13.4282	−1.47394	−0.736970	+	0.675925i	$0.763744\pi$
$84$	0	0
$85$	−0.0797159	−0.00864640
$86$	0	0
$87$	1.29094	0.138403
$88$	0	0
$89$	−10.0457	−1.06484	−0.532420	−	0.846481i	$-0.678717\pi$
$89$	−10.0457	−1.06484	−0.532420	+	0.846481i	$0.678717\pi$
$90$	0	0
$91$	−2.78539	−0.291988
$92$	0	0
$93$	−1.21322	−0.125805
$94$	0	0
$95$	−0.140763	−0.0144420
$96$	0	0
$97$	11.8338	1.20154	0.600770	−	0.799422i	$-0.294861\pi$
$97$	11.8338	1.20154	0.600770	+	0.799422i	$0.294861\pi$
$98$	0	0
$99$	−0.863742	−0.0868094
$100$	0	0
$101$	−15.7416	−1.56635	−0.783175	−	0.621801i	$-0.786401\pi$
$101$	−15.7416	−1.56635	−0.783175	+	0.621801i	$0.786401\pi$
$102$	0	0
$103$	−11.4296	−1.12619	−0.563094	−	0.826393i	$-0.690389\pi$
$103$	−11.4296	−1.12619	−0.563094	+	0.826393i	$0.690389\pi$
$104$	0	0
$105$	0.0223113	0.00217736
$106$	0	0
$107$	−8.44980	−0.816873	−0.408437	−	0.912787i	$-0.633926\pi$
$107$	−8.44980	−0.816873	−0.408437	+	0.912787i	$0.633926\pi$
$108$	0	0
$109$	7.81853	0.748879	0.374440	−	0.927251i	$-0.377835\pi$
$109$	7.81853	0.748879	0.374440	+	0.927251i	$0.377835\pi$
$110$	0	0
$111$	7.14619	0.678287
$112$	0	0
$113$	−2.40154	−0.225918	−0.112959	−	0.993600i	$-0.536033\pi$
$113$	−2.40154	−0.225918	−0.112959	+	0.993600i	$0.536033\pi$
$114$	0	0
$115$	0.288034	0.0268593
$116$	0	0
$117$	−4.36921	−0.403933
$118$	0	0
$119$	−1.45207	−0.133111
$120$	0	0
$121$	−10.2539	−0.932177
$122$	0	0
$123$	2.42495	0.218651
$124$	0	0
$125$	−0.349935	−0.0312992
$126$	0	0
$127$	−7.45717	−0.661717	−0.330858	−	0.943680i	$-0.607338\pi$
$127$	−7.45717	−0.661717	−0.330858	+	0.943680i	$0.607338\pi$
$128$	0	0
$129$	0.00158582	0.000139624 0
$130$	0	0
$131$	−0.0329360	−0.00287763	−0.00143882	−	0.999999i	$-0.500458\pi$
$131$	−0.0329360	−0.00287763	−0.00143882	+	0.999999i	$0.500458\pi$
$132$	0	0
$133$	−2.56408	−0.222334
$134$	0	0
$135$	0.0349978	0.00301213
$136$	0	0
$137$	−13.7940	−1.17850	−0.589252	−	0.807949i	$-0.700577\pi$
$137$	−13.7940	−1.17850	−0.589252	+	0.807949i	$0.700577\pi$
$138$	0	0
$139$	6.22618	0.528098	0.264049	−	0.964509i	$-0.414942\pi$
$139$	6.22618	0.528098	0.264049	+	0.964509i	$0.414942\pi$
$140$	0	0
$141$	−3.98820	−0.335867
$142$	0	0
$143$	3.77387	0.315587
$144$	0	0
$145$	0.0451800	0.00375199
$146$	0	0
$147$	−6.59359	−0.543830
$148$	0	0
$149$	13.5572	1.11065	0.555326	−	0.831633i	$-0.312594\pi$
$149$	13.5572	1.11065	0.555326	+	0.831633i	$0.312594\pi$
$150$	0	0
$151$	4.23542	0.344674	0.172337	−	0.985038i	$-0.444868\pi$
$151$	4.23542	0.344674	0.172337	+	0.985038i	$0.444868\pi$
$152$	0	0
$153$	−2.27774	−0.184144
$154$	0	0
$155$	−0.0424601	−0.00341048
$156$	0	0
$157$	9.88889	0.789219	0.394610	−	0.918849i	$-0.370880\pi$
$157$	9.88889	0.789219	0.394610	+	0.918849i	$0.370880\pi$
$158$	0	0
$159$	7.67263	0.608479
$160$	0	0
$161$	5.24671	0.413498
$162$	0	0
$163$	7.43335	0.582225	0.291112	−	0.956689i	$-0.405975\pi$
$163$	7.43335	0.582225	0.291112	+	0.956689i	$0.405975\pi$
$164$	0	0
$165$	−0.0302291	−0.00235333
$166$	0	0
$167$	−4.80704	−0.371980	−0.185990	−	0.982552i	$-0.559549\pi$
$167$	−4.80704	−0.371980	−0.185990	+	0.982552i	$0.559549\pi$
$168$	0	0
$169$	6.08997	0.468459
$170$	0	0
$171$	−4.02205	−0.307574
$172$	0	0
$173$	−20.4319	−1.55341	−0.776704	−	0.629866i	$-0.783110\pi$
$173$	−20.4319	−1.55341	−0.776704	+	0.629866i	$0.783110\pi$
$174$	0	0
$175$	−3.18674	−0.240895
$176$	0	0
$177$	−1.59240	−0.119692
$178$	0	0
$179$	−1.81895	−0.135955	−0.0679774	−	0.997687i	$-0.521655\pi$
$179$	−1.81895	−0.135955	−0.0679774	+	0.997687i	$0.521655\pi$
$180$	0	0
$181$	−5.92712	−0.440559	−0.220280	−	0.975437i	$-0.570697\pi$
$181$	−5.92712	−0.440559	−0.220280	+	0.975437i	$0.570697\pi$
$182$	0	0
$183$	−3.50074	−0.258782
$184$	0	0
$185$	0.250101	0.0183878
$186$	0	0
$187$	1.96738	0.143869
$188$	0	0
$189$	0.637505	0.0463717
$190$	0	0
$191$	0.852539	0.0616876	0.0308438	−	0.999524i	$-0.490181\pi$
$191$	0.852539	0.0616876	0.0308438	+	0.999524i	$0.490181\pi$
$192$	0	0
$193$	−0.273987	−0.0197220	−0.00986099	−	0.999951i	$-0.503139\pi$
$193$	−0.273987	−0.0197220	−0.00986099	+	0.999951i	$0.503139\pi$
$194$	0	0
$195$	−0.152913	−0.0109503
$196$	0	0
$197$	−5.07126	−0.361313	−0.180656	−	0.983546i	$-0.557822\pi$
$197$	−5.07126	−0.361313	−0.180656	+	0.983546i	$0.557822\pi$
$198$	0	0
$199$	4.29664	0.304581	0.152290	−	0.988336i	$-0.451335\pi$
$199$	4.29664	0.304581	0.152290	+	0.988336i	$0.451335\pi$
$200$	0	0
$201$	−15.8723	−1.11955
$202$	0	0
$203$	0.822979	0.0577618
$204$	0	0
$205$	0.0848680	0.00592744
$206$	0	0
$207$	8.23006	0.572029
$208$	0	0
$209$	3.47401	0.240303
$210$	0	0
$211$	19.7928	1.36259	0.681295	−	0.732009i	$-0.261417\pi$
$211$	19.7928	1.36259	0.681295	+	0.732009i	$0.261417\pi$
$212$	0	0
$213$	−12.8739	−0.882108
$214$	0	0
$215$	5.55004e−5 0	3.78509e−6 0
$216$	0	0
$217$	−0.773435	−0.0525042
$218$	0	0
$219$	−2.85083	−0.192642
$220$	0	0
$221$	9.95192	0.669438
$222$	0	0
$223$	−0.948794	−0.0635360	−0.0317680	−	0.999495i	$-0.510114\pi$
$223$	−0.948794	−0.0635360	−0.0317680	+	0.999495i	$0.510114\pi$
$224$	0	0
$225$	−4.99878	−0.333252
$226$	0	0
$227$	15.3021	1.01563	0.507817	−	0.861465i	$-0.330453\pi$
$227$	15.3021	1.01563	0.507817	+	0.861465i	$0.330453\pi$
$228$	0	0
$229$	−14.9304	−0.986626	−0.493313	−	0.869852i	$-0.664214\pi$
$229$	−14.9304	−0.986626	−0.493313	+	0.869852i	$0.664214\pi$
$230$	0	0
$231$	−0.550640	−0.0362294
$232$	0	0
$233$	1.04199	0.0682630	0.0341315	−	0.999417i	$-0.489133\pi$
$233$	1.04199	0.0682630	0.0341315	+	0.999417i	$0.489133\pi$
$234$	0	0
$235$	−0.139578	−0.00910509
$236$	0	0
$237$	7.20069	0.467735
$238$	0	0
$239$	−4.80544	−0.310838	−0.155419	−	0.987849i	$-0.549673\pi$
$239$	−4.80544	−0.310838	−0.155419	+	0.987849i	$0.549673\pi$
$240$	0	0
$241$	24.4322	1.57381	0.786907	−	0.617071i	$-0.211681\pi$
$241$	24.4322	1.57381	0.786907	+	0.617071i	$0.211681\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.230761	−0.0147428
$246$	0	0
$247$	17.5732	1.11815
$248$	0	0
$249$	−13.4282	−0.850980
$250$	0	0
$251$	−4.80461	−0.303264	−0.151632	−	0.988437i	$-0.548453\pi$
$251$	−4.80461	−0.303264	−0.151632	+	0.988437i	$0.548453\pi$
$252$	0	0
$253$	−7.10865	−0.446917
$254$	0	0
$255$	−0.0797159	−0.00499200
$256$	0	0
$257$	−18.2299	−1.13715	−0.568576	−	0.822631i	$-0.692505\pi$
$257$	−18.2299	−1.13715	−0.568576	+	0.822631i	$0.692505\pi$
$258$	0	0
$259$	4.55573	0.283079
$260$	0	0
$261$	1.29094	0.0799070
$262$	0	0
$263$	−27.2527	−1.68047	−0.840235	−	0.542222i	$-0.817583\pi$
$263$	−27.2527	−1.68047	−0.840235	+	0.542222i	$0.817583\pi$
$264$	0	0
$265$	0.268525	0.0164954
$266$	0	0
$267$	−10.0457	−0.614785
$268$	0	0
$269$	15.4718	0.943335	0.471667	−	0.881777i	$-0.343652\pi$
$269$	15.4718	0.943335	0.471667	+	0.881777i	$0.343652\pi$
$270$	0	0
$271$	12.3488	0.750138	0.375069	−	0.926997i	$-0.377619\pi$
$271$	12.3488	0.750138	0.375069	+	0.926997i	$0.377619\pi$
$272$	0	0
$273$	−2.78539	−0.168580
$274$	0	0
$275$	4.31765	0.260364
$276$	0	0
$277$	13.0131	0.781883	0.390941	−	0.920416i	$-0.372150\pi$
$277$	13.0131	0.781883	0.390941	+	0.920416i	$0.372150\pi$
$278$	0	0
$279$	−1.21322	−0.0726337
$280$	0	0
$281$	14.6624	0.874683	0.437341	−	0.899296i	$-0.355920\pi$
$281$	14.6624	0.874683	0.437341	+	0.899296i	$0.355920\pi$
$282$	0	0
$283$	4.37195	0.259886	0.129943	−	0.991521i	$-0.458521\pi$
$283$	4.37195	0.259886	0.129943	+	0.991521i	$0.458521\pi$
$284$	0	0
$285$	−0.140763	−0.00833808
$286$	0	0
$287$	1.54592	0.0912527
$288$	0	0
$289$	−11.8119	−0.694818
$290$	0	0
$291$	11.8338	0.693710
$292$	0	0
$293$	−23.6309	−1.38053	−0.690267	−	0.723555i	$-0.742507\pi$
$293$	−23.6309	−1.38053	−0.690267	+	0.723555i	$0.742507\pi$
$294$	0	0
$295$	−0.0557306	−0.00324476
$296$	0	0
$297$	−0.863742	−0.0501194
$298$	0	0
$299$	−35.9588	−2.07955
$300$	0	0
$301$	0.00101097	5.82714e−5 0
$302$	0	0
$303$	−15.7416	−0.904333
$304$	0	0
$305$	−0.122518	−0.00701538
$306$	0	0
$307$	16.8530	0.961851	0.480926	−	0.876761i	$-0.340301\pi$
$307$	16.8530	0.961851	0.480926	+	0.876761i	$0.340301\pi$
$308$	0	0
$309$	−11.4296	−0.650205
$310$	0	0
$311$	−8.09954	−0.459283	−0.229641	−	0.973275i	$-0.573755\pi$
$311$	−8.09954	−0.459283	−0.229641	+	0.973275i	$0.573755\pi$
$312$	0	0
$313$	−27.5042	−1.55463	−0.777315	−	0.629112i	$-0.783419\pi$
$313$	−27.5042	−1.55463	−0.777315	+	0.629112i	$0.783419\pi$
$314$	0	0
$315$	0.0223113	0.00125710
$316$	0	0
$317$	−22.0581	−1.23891	−0.619454	−	0.785033i	$-0.712646\pi$
$317$	−22.0581	−1.23891	−0.619454	+	0.785033i	$0.712646\pi$
$318$	0	0
$319$	−1.11504	−0.0624301
$320$	0	0
$321$	−8.44980	−0.471622
$322$	0	0
$323$	9.16118	0.509742
$324$	0	0
$325$	21.8407	1.21150
$326$	0	0
$327$	7.81853	0.432366
$328$	0	0
$329$	−2.54250	−0.140173
$330$	0	0
$331$	22.3009	1.22577	0.612883	−	0.790174i	$-0.290010\pi$
$331$	22.3009	1.22577	0.612883	+	0.790174i	$0.290010\pi$
$332$	0	0
$333$	7.14619	0.391609
$334$	0	0
$335$	−0.555497	−0.0303500
$336$	0	0
$337$	14.1121	0.768735	0.384368	−	0.923180i	$-0.374420\pi$
$337$	14.1121	0.768735	0.384368	+	0.923180i	$0.374420\pi$
$338$	0	0
$339$	−2.40154	−0.130434
$340$	0	0
$341$	1.04791	0.0567476
$342$	0	0
$343$	−8.66598	−0.467919
$344$	0	0
$345$	0.288034	0.0155072
$346$	0	0
$347$	15.3510	0.824087	0.412043	−	0.911164i	$-0.364815\pi$
$347$	15.3510	0.824087	0.412043	+	0.911164i	$0.364815\pi$
$348$	0	0
$349$	−10.2174	−0.546925	−0.273462	−	0.961883i	$-0.588169\pi$
$349$	−10.2174	−0.546925	−0.273462	+	0.961883i	$0.588169\pi$
$350$	0	0
$351$	−4.36921	−0.233211
$352$	0	0
$353$	15.3114	0.814944	0.407472	−	0.913218i	$-0.366410\pi$
$353$	15.3114	0.814944	0.407472	+	0.913218i	$0.366410\pi$
$354$	0	0
$355$	−0.450560	−0.0239132
$356$	0	0
$357$	−1.45207	−0.0768517
$358$	0	0
$359$	−12.8431	−0.677833	−0.338917	−	0.940816i	$-0.610060\pi$
$359$	−12.8431	−0.677833	−0.338917	+	0.940816i	$0.610060\pi$
$360$	0	0
$361$	−2.82311	−0.148585
$362$	0	0
$363$	−10.2539	−0.538193
$364$	0	0
$365$	−0.0997730	−0.00522236
$366$	0	0
$367$	−24.5432	−1.28114	−0.640571	−	0.767899i	$-0.721302\pi$
$367$	−24.5432	−1.28114	−0.640571	+	0.767899i	$0.721302\pi$
$368$	0	0
$369$	2.42495	0.126238
$370$	0	0
$371$	4.89134	0.253946
$372$	0	0
$373$	−11.9642	−0.619485	−0.309743	−	0.950820i	$-0.600243\pi$
$373$	−11.9642	−0.619485	−0.309743	+	0.950820i	$0.600243\pi$
$374$	0	0
$375$	−0.349935	−0.0180706
$376$	0	0
$377$	−5.64037	−0.290494
$378$	0	0
$379$	−13.6386	−0.700568	−0.350284	−	0.936643i	$-0.613915\pi$
$379$	−13.6386	−0.700568	−0.350284	+	0.936643i	$0.613915\pi$
$380$	0	0
$381$	−7.45717	−0.382042
$382$	0	0
$383$	−3.00991	−0.153799	−0.0768996	−	0.997039i	$-0.524502\pi$
$383$	−3.00991	−0.153799	−0.0768996	+	0.997039i	$0.524502\pi$
$384$	0	0
$385$	−0.0192712	−0.000982151 0
$386$	0	0
$387$	0.00158582	8.06120e−5 0
$388$	0	0
$389$	20.9575	1.06259	0.531295	−	0.847187i	$-0.321706\pi$
$389$	20.9575	1.06259	0.531295	+	0.847187i	$0.321706\pi$
$390$	0	0
$391$	−18.7459	−0.948023
$392$	0	0
$393$	−0.0329360	−0.00166140
$394$	0	0
$395$	0.252008	0.0126799
$396$	0	0
$397$	7.01282	0.351963	0.175982	−	0.984393i	$-0.443690\pi$
$397$	7.01282	0.351963	0.175982	+	0.984393i	$0.443690\pi$
$398$	0	0
$399$	−2.56408	−0.128364
$400$	0	0
$401$	−24.2870	−1.21284	−0.606418	−	0.795146i	$-0.707394\pi$
$401$	−24.2870	−1.21284	−0.606418	+	0.795146i	$0.707394\pi$
$402$	0	0
$403$	5.30082	0.264053
$404$	0	0
$405$	0.0349978	0.00173905
$406$	0	0
$407$	−6.17247	−0.305958
$408$	0	0
$409$	−19.3926	−0.958901	−0.479450	−	0.877569i	$-0.659164\pi$
$409$	−19.3926	−0.958901	−0.479450	+	0.877569i	$0.659164\pi$
$410$	0	0
$411$	−13.7940	−0.680410
$412$	0	0
$413$	−1.01517	−0.0499530
$414$	0	0
$415$	−0.469959	−0.0230694
$416$	0	0
$417$	6.22618	0.304897
$418$	0	0
$419$	18.6325	0.910256	0.455128	−	0.890426i	$-0.349594\pi$
$419$	18.6325	0.910256	0.455128	+	0.890426i	$0.349594\pi$
$420$	0	0
$421$	−22.8703	−1.11463	−0.557315	−	0.830301i	$-0.688168\pi$
$421$	−22.8703	−1.11463	−0.557315	+	0.830301i	$0.688168\pi$
$422$	0	0
$423$	−3.98820	−0.193913
$424$	0	0
$425$	11.3859	0.552298
$426$	0	0
$427$	−2.23174	−0.108001
$428$	0	0
$429$	3.77387	0.182204
$430$	0	0
$431$	25.2271	1.21515	0.607573	−	0.794264i	$-0.292143\pi$
$431$	25.2271	1.21515	0.607573	+	0.794264i	$0.292143\pi$
$432$	0	0
$433$	−5.95961	−0.286400	−0.143200	−	0.989694i	$-0.545739\pi$
$433$	−5.95961	−0.286400	−0.143200	+	0.989694i	$0.545739\pi$
$434$	0	0
$435$	0.0451800	0.00216621
$436$	0	0
$437$	−33.1017	−1.58347
$438$	0	0
$439$	−0.672432	−0.0320934	−0.0160467	−	0.999871i	$-0.505108\pi$
$439$	−0.672432	−0.0320934	−0.0160467	+	0.999871i	$0.505108\pi$
$440$	0	0
$441$	−6.59359	−0.313980
$442$	0	0
$443$	32.4879	1.54354	0.771772	−	0.635899i	$-0.219370\pi$
$443$	32.4879	1.54354	0.771772	+	0.635899i	$0.219370\pi$
$444$	0	0
$445$	−0.351577	−0.0166663
$446$	0	0
$447$	13.5572	0.641235
$448$	0	0
$449$	2.54165	0.119948	0.0599740	−	0.998200i	$-0.480898\pi$
$449$	2.54165	0.119948	0.0599740	+	0.998200i	$0.480898\pi$
$450$	0	0
$451$	−2.09453	−0.0986277
$452$	0	0
$453$	4.23542	0.198998
$454$	0	0
$455$	−0.0974826	−0.00457005
$456$	0	0
$457$	14.7451	0.689747	0.344873	−	0.938649i	$-0.387922\pi$
$457$	14.7451	0.689747	0.344873	+	0.938649i	$0.387922\pi$
$458$	0	0
$459$	−2.27774	−0.106316
$460$	0	0
$461$	11.0554	0.514900	0.257450	−	0.966292i	$-0.417118\pi$
$461$	11.0554	0.514900	0.257450	+	0.966292i	$0.417118\pi$
$462$	0	0
$463$	9.73323	0.452341	0.226171	−	0.974088i	$-0.427379\pi$
$463$	9.73323	0.452341	0.226171	+	0.974088i	$0.427379\pi$
$464$	0	0
$465$	−0.0424601	−0.00196904
$466$	0	0
$467$	−19.1438	−0.885867	−0.442934	−	0.896554i	$-0.646062\pi$
$467$	−19.1438	−0.885867	−0.442934	+	0.896554i	$0.646062\pi$
$468$	0	0
$469$	−10.1187	−0.467238
$470$	0	0
$471$	9.88889	0.455656
$472$	0	0
$473$	−0.00136974	−6.29809e−5 0
$474$	0	0
$475$	20.1053	0.922496
$476$	0	0
$477$	7.67263	0.351305
$478$	0	0
$479$	−6.43999	−0.294251	−0.147125	−	0.989118i	$-0.547002\pi$
$479$	−6.43999	−0.294251	−0.147125	+	0.989118i	$0.547002\pi$
$480$	0	0
$481$	−31.2232	−1.42366
$482$	0	0
$483$	5.24671	0.238733
$484$	0	0
$485$	0.414157	0.0188059
$486$	0	0
$487$	−37.0468	−1.67875	−0.839376	−	0.543552i	$-0.817079\pi$
$487$	−37.0468	−1.67875	−0.839376	+	0.543552i	$0.817079\pi$
$488$	0	0
$489$	7.43335	0.336148
$490$	0	0
$491$	−0.353508	−0.0159536	−0.00797680	−	0.999968i	$-0.502539\pi$
$491$	−0.353508	−0.0159536	−0.00797680	+	0.999968i	$0.502539\pi$
$492$	0	0
$493$	−2.94042	−0.132430
$494$	0	0
$495$	−0.0302291	−0.00135870
$496$	0	0
$497$	−8.20720	−0.368143
$498$	0	0
$499$	3.01632	0.135029	0.0675144	−	0.997718i	$-0.478493\pi$
$499$	3.01632	0.135029	0.0675144	+	0.997718i	$0.478493\pi$
$500$	0	0
$501$	−4.80704	−0.214763
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−0.550922	−0.0245157
$506$	0	0
$507$	6.08997	0.270465
$508$	0	0
$509$	22.6447	1.00371	0.501855	−	0.864952i	$-0.332651\pi$
$509$	22.6447	1.00371	0.501855	+	0.864952i	$0.332651\pi$
$510$	0	0
$511$	−1.81742	−0.0803980
$512$	0	0
$513$	−4.02205	−0.177578
$514$	0	0
$515$	−0.400009	−0.0176265
$516$	0	0
$517$	3.44478	0.151501
$518$	0	0
$519$	−20.4319	−0.896860
$520$	0	0
$521$	−36.5888	−1.60298	−0.801492	−	0.598006i	$-0.795960\pi$
$521$	−36.5888	−1.60298	−0.801492	+	0.598006i	$0.795960\pi$
$522$	0	0
$523$	5.01297	0.219202	0.109601	−	0.993976i	$-0.465043\pi$
$523$	5.01297	0.219202	0.109601	+	0.993976i	$0.465043\pi$
$524$	0	0
$525$	−3.18674	−0.139081
$526$	0	0
$527$	2.76341	0.120376
$528$	0	0
$529$	44.7339	1.94495
$530$	0	0
$531$	−1.59240	−0.0691044
$532$	0	0
$533$	−10.5951	−0.458925
$534$	0	0
$535$	−0.295725	−0.0127853
$536$	0	0
$537$	−1.81895	−0.0784936
$538$	0	0
$539$	5.69516	0.245308
$540$	0	0
$541$	23.3051	1.00197	0.500983	−	0.865457i	$-0.332972\pi$
$541$	23.3051	1.00197	0.500983	+	0.865457i	$0.332972\pi$
$542$	0	0
$543$	−5.92712	−0.254357
$544$	0	0
$545$	0.273631	0.0117211
$546$	0	0
$547$	10.8082	0.462126	0.231063	−	0.972939i	$-0.425780\pi$
$547$	10.8082	0.462126	0.231063	+	0.972939i	$0.425780\pi$
$548$	0	0
$549$	−3.50074	−0.149408
$550$	0	0
$551$	−5.19221	−0.221196
$552$	0	0
$553$	4.59047	0.195207
$554$	0	0
$555$	0.250101	0.0106162
$556$	0	0
$557$	28.5876	1.21130	0.605648	−	0.795733i	$-0.292914\pi$
$557$	28.5876	1.21130	0.605648	+	0.795733i	$0.292914\pi$
$558$	0	0
$559$	−0.00692879	−0.000293057 0
$560$	0	0
$561$	1.96738	0.0830629
$562$	0	0
$563$	8.89314	0.374801	0.187401	−	0.982284i	$-0.439994\pi$
$563$	8.89314	0.374801	0.187401	+	0.982284i	$0.439994\pi$
$564$	0	0
$565$	−0.0840487	−0.00353596
$566$	0	0
$567$	0.637505	0.0267727
$568$	0	0
$569$	−16.6552	−0.698222	−0.349111	−	0.937081i	$-0.613516\pi$
$569$	−16.6552	−0.698222	−0.349111	+	0.937081i	$0.613516\pi$
$570$	0	0
$571$	−4.35772	−0.182365	−0.0911825	−	0.995834i	$-0.529065\pi$
$571$	−4.35772	−0.182365	−0.0911825	+	0.995834i	$0.529065\pi$
$572$	0	0
$573$	0.852539	0.0356153
$574$	0	0
$575$	−41.1402	−1.71567
$576$	0	0
$577$	3.48423	0.145050	0.0725251	−	0.997367i	$-0.476894\pi$
$577$	3.48423	0.145050	0.0725251	+	0.997367i	$0.476894\pi$
$578$	0	0
$579$	−0.273987	−0.0113865
$580$	0	0
$581$	−8.56057	−0.355152
$582$	0	0
$583$	−6.62717	−0.274469
$584$	0	0
$585$	−0.152913	−0.00632216
$586$	0	0
$587$	−8.60458	−0.355149	−0.177575	−	0.984107i	$-0.556825\pi$
$587$	−8.60458	−0.355149	−0.177575	+	0.984107i	$0.556825\pi$
$588$	0	0
$589$	4.87964	0.201062
$590$	0	0
$591$	−5.07126	−0.208604
$592$	0	0
$593$	30.3841	1.24773	0.623863	−	0.781533i	$-0.285562\pi$
$593$	30.3841	1.24773	0.623863	+	0.781533i	$0.285562\pi$
$594$	0	0
$595$	−0.0508193	−0.00208339
$596$	0	0
$597$	4.29664	0.175850
$598$	0	0
$599$	23.5843	0.963626	0.481813	−	0.876274i	$-0.339978\pi$
$599$	23.5843	0.963626	0.481813	+	0.876274i	$0.339978\pi$
$600$	0	0
$601$	10.6263	0.433455	0.216728	−	0.976232i	$-0.430462\pi$
$601$	10.6263	0.433455	0.216728	+	0.976232i	$0.430462\pi$
$602$	0	0
$603$	−15.8723	−0.646371
$604$	0	0
$605$	−0.358866	−0.0145900
$606$	0	0
$607$	9.78081	0.396991	0.198495	−	0.980102i	$-0.436395\pi$
$607$	9.78081	0.396991	0.198495	+	0.980102i	$0.436395\pi$
$608$	0	0
$609$	0.822979	0.0333488
$610$	0	0
$611$	17.4253	0.704952
$612$	0	0
$613$	24.6552	0.995813	0.497906	−	0.867231i	$-0.334102\pi$
$613$	24.6552	0.995813	0.497906	+	0.867231i	$0.334102\pi$
$614$	0	0
$615$	0.0848680	0.00342221
$616$	0	0
$617$	−17.2052	−0.692657	−0.346328	−	0.938113i	$-0.612572\pi$
$617$	−17.2052	−0.692657	−0.346328	+	0.938113i	$0.612572\pi$
$618$	0	0
$619$	18.2401	0.733133	0.366567	−	0.930392i	$-0.380533\pi$
$619$	18.2401	0.733133	0.366567	+	0.930392i	$0.380533\pi$
$620$	0	0
$621$	8.23006	0.330261
$622$	0	0
$623$	−6.40417	−0.256578
$624$	0	0
$625$	24.9816	0.999265
$626$	0	0
$627$	3.47401	0.138739
$628$	0	0
$629$	−16.2772	−0.649013
$630$	0	0
$631$	25.3591	1.00953	0.504766	−	0.863256i	$-0.331579\pi$
$631$	25.3591	1.00953	0.504766	+	0.863256i	$0.331579\pi$
$632$	0	0
$633$	19.7928	0.786692
$634$	0	0
$635$	−0.260985	−0.0103569
$636$	0	0
$637$	28.8087	1.14144
$638$	0	0
$639$	−12.8739	−0.509285
$640$	0	0
$641$	7.75302	0.306226	0.153113	−	0.988209i	$-0.451070\pi$
$641$	7.75302	0.306226	0.153113	+	0.988209i	$0.451070\pi$
$642$	0	0
$643$	22.4658	0.885966	0.442983	−	0.896530i	$-0.353920\pi$
$643$	22.4658	0.885966	0.442983	+	0.896530i	$0.353920\pi$
$644$	0	0
$645$	5.55004e−5 0	2.18532e−6 0
$646$	0	0
$647$	5.50543	0.216441	0.108220	−	0.994127i	$-0.465485\pi$
$647$	5.50543	0.216441	0.108220	+	0.994127i	$0.465485\pi$
$648$	0	0
$649$	1.37543	0.0539902
$650$	0	0
$651$	−0.773435	−0.0303133
$652$	0	0
$653$	8.38115	0.327980	0.163990	−	0.986462i	$-0.447564\pi$
$653$	8.38115	0.327980	0.163990	+	0.986462i	$0.447564\pi$
$654$	0	0
$655$	−0.00115269	−4.50393e−5 0
$656$	0	0
$657$	−2.85083	−0.111222
$658$	0	0
$659$	24.1879	0.942225	0.471112	−	0.882073i	$-0.343853\pi$
$659$	24.1879	0.942225	0.471112	+	0.882073i	$0.343853\pi$
$660$	0	0
$661$	13.8859	0.540098	0.270049	−	0.962847i	$-0.412960\pi$
$661$	13.8859	0.540098	0.270049	+	0.962847i	$0.412960\pi$
$662$	0	0
$663$	9.95192	0.386500
$664$	0	0
$665$	−0.0897371	−0.00347985
$666$	0	0
$667$	10.6245	0.411382
$668$	0	0
$669$	−0.948794	−0.0366825
$670$	0	0
$671$	3.02374	0.116730
$672$	0	0
$673$	−36.6544	−1.41292	−0.706461	−	0.707752i	$-0.749709\pi$
$673$	−36.6544	−1.41292	−0.706461	+	0.707752i	$0.749709\pi$
$674$	0	0
$675$	−4.99878	−0.192403
$676$	0	0
$677$	−32.4036	−1.24537	−0.622685	−	0.782473i	$-0.713958\pi$
$677$	−32.4036	−1.24537	−0.622685	+	0.782473i	$0.713958\pi$
$678$	0	0
$679$	7.54410	0.289516
$680$	0	0
$681$	15.3021	0.586377
$682$	0	0
$683$	31.3861	1.20095	0.600477	−	0.799642i	$-0.294977\pi$
$683$	31.3861	1.20095	0.600477	+	0.799642i	$0.294977\pi$
$684$	0	0
$685$	−0.482761	−0.0184453
$686$	0	0
$687$	−14.9304	−0.569629
$688$	0	0
$689$	−33.5233	−1.27714
$690$	0	0
$691$	−4.62328	−0.175878	−0.0879389	−	0.996126i	$-0.528028\pi$
$691$	−4.62328	−0.175878	−0.0879389	+	0.996126i	$0.528028\pi$
$692$	0	0
$693$	−0.550640	−0.0209171
$694$	0	0
$695$	0.217903	0.00826552
$696$	0	0
$697$	−5.52341	−0.209214
$698$	0	0
$699$	1.04199	0.0394117
$700$	0	0
$701$	−18.8324	−0.711288	−0.355644	−	0.934621i	$-0.615738\pi$
$701$	−18.8324	−0.711288	−0.355644	+	0.934621i	$0.615738\pi$
$702$	0	0
$703$	−28.7423	−1.08404
$704$	0	0
$705$	−0.139578	−0.00525683
$706$	0	0
$707$	−10.0354	−0.377419
$708$	0	0
$709$	−27.1420	−1.01934	−0.509671	−	0.860370i	$-0.670233\pi$
$709$	−27.1420	−1.01934	−0.509671	+	0.860370i	$0.670233\pi$
$710$	0	0
$711$	7.20069	0.270047
$712$	0	0
$713$	−9.98490	−0.373937
$714$	0	0
$715$	0.132077	0.00493940
$716$	0	0
$717$	−4.80544	−0.179462
$718$	0	0
$719$	−48.9125	−1.82413	−0.912065	−	0.410046i	$-0.865513\pi$
$719$	−48.9125	−1.82413	−0.912065	+	0.410046i	$0.865513\pi$
$720$	0	0
$721$	−7.28640	−0.271360
$722$	0	0
$723$	24.4322	0.908642
$724$	0	0
$725$	−6.45310	−0.239662
$726$	0	0
$727$	47.3184	1.75494	0.877471	−	0.479630i	$-0.159229\pi$
$727$	47.3184	1.75494	0.877471	+	0.479630i	$0.159229\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.00361209	−0.000133598 0
$732$	0	0
$733$	22.1527	0.818230	0.409115	−	0.912483i	$-0.365837\pi$
$733$	22.1527	0.818230	0.409115	+	0.912483i	$0.365837\pi$
$734$	0	0
$735$	−0.230761	−0.00851175
$736$	0	0
$737$	13.7096	0.505000
$738$	0	0
$739$	23.6128	0.868612	0.434306	−	0.900765i	$-0.356994\pi$
$739$	23.6128	0.868612	0.434306	+	0.900765i	$0.356994\pi$
$740$	0	0
$741$	17.5732	0.645566
$742$	0	0
$743$	18.2444	0.669321	0.334661	−	0.942339i	$-0.391378\pi$
$743$	18.2444	0.669321	0.334661	+	0.942339i	$0.391378\pi$
$744$	0	0
$745$	0.474473	0.0173833
$746$	0	0
$747$	−13.4282	−0.491313
$748$	0	0
$749$	−5.38679	−0.196829
$750$	0	0
$751$	−38.7717	−1.41480	−0.707399	−	0.706815i	$-0.750132\pi$
$751$	−38.7717	−1.41480	−0.707399	+	0.706815i	$0.750132\pi$
$752$	0	0
$753$	−4.80461	−0.175090
$754$	0	0
$755$	0.148231	0.00539466
$756$	0	0
$757$	46.9590	1.70675	0.853377	−	0.521293i	$-0.174550\pi$
$757$	46.9590	1.70675	0.853377	+	0.521293i	$0.174550\pi$
$758$	0	0
$759$	−7.10865	−0.258028
$760$	0	0
$761$	−11.4071	−0.413507	−0.206754	−	0.978393i	$-0.566290\pi$
$761$	−11.4071	−0.413507	−0.206754	+	0.978393i	$0.566290\pi$
$762$	0	0
$763$	4.98435	0.180446
$764$	0	0
$765$	−0.0797159	−0.00288213
$766$	0	0
$767$	6.95754	0.251222
$768$	0	0
$769$	21.6809	0.781833	0.390917	−	0.920426i	$-0.372158\pi$
$769$	21.6809	0.781833	0.390917	+	0.920426i	$0.372158\pi$
$770$	0	0
$771$	−18.2299	−0.656535
$772$	0	0
$773$	−8.81854	−0.317181	−0.158590	−	0.987344i	$-0.550695\pi$
$773$	−8.81854	−0.317181	−0.158590	+	0.987344i	$0.550695\pi$
$774$	0	0
$775$	6.06463	0.217848
$776$	0	0
$777$	4.55573	0.163436
$778$	0	0
$779$	−9.75328	−0.349448
$780$	0	0
$781$	11.1198	0.397896
$782$	0	0
$783$	1.29094	0.0461343
$784$	0	0
$785$	0.346089	0.0123525
$786$	0	0
$787$	−9.74371	−0.347326	−0.173663	−	0.984805i	$-0.555560\pi$
$787$	−9.74371	−0.347326	−0.173663	+	0.984805i	$0.555560\pi$
$788$	0	0
$789$	−27.2527	−0.970220
$790$	0	0
$791$	−1.53100	−0.0544359
$792$	0	0
$793$	15.2955	0.543158
$794$	0	0
$795$	0.268525	0.00952360
$796$	0	0
$797$	−40.7895	−1.44484	−0.722419	−	0.691456i	$-0.756970\pi$
$797$	−40.7895	−1.44484	−0.722419	+	0.691456i	$0.756970\pi$
$798$	0	0
$799$	9.08409	0.321372
$800$	0	0
$801$	−10.0457	−0.354946
$802$	0	0
$803$	2.46239	0.0868957
$804$	0	0
$805$	0.183623	0.00647187
$806$	0	0
$807$	15.4718	0.544635
$808$	0	0
$809$	28.4540	1.00039	0.500195	−	0.865913i	$-0.333262\pi$
$809$	28.4540	1.00039	0.500195	+	0.865913i	$0.333262\pi$
$810$	0	0
$811$	22.2541	0.781447	0.390723	−	0.920508i	$-0.372225\pi$
$811$	22.2541	0.781447	0.390723	+	0.920508i	$0.372225\pi$
$812$	0	0
$813$	12.3488	0.433092
$814$	0	0
$815$	0.260151	0.00911269
$816$	0	0
$817$	−0.00637826	−0.000223147 0
$818$	0	0
$819$	−2.78539	−0.0973294
$820$	0	0
$821$	−44.4240	−1.55041	−0.775203	−	0.631712i	$-0.782353\pi$
$821$	−44.4240	−1.55041	−0.775203	+	0.631712i	$0.782353\pi$
$822$	0	0
$823$	−47.1094	−1.64213	−0.821065	−	0.570835i	$-0.806619\pi$
$823$	−47.1094	−1.64213	−0.821065	+	0.570835i	$0.806619\pi$
$824$	0	0
$825$	4.31765	0.150321
$826$	0	0
$827$	−33.0786	−1.15026	−0.575128	−	0.818063i	$-0.695048\pi$
$827$	−33.0786	−1.15026	−0.575128	+	0.818063i	$0.695048\pi$
$828$	0	0
$829$	−40.0005	−1.38927	−0.694637	−	0.719361i	$-0.744435\pi$
$829$	−40.0005	−1.38927	−0.694637	+	0.719361i	$0.744435\pi$
$830$	0	0
$831$	13.0131	0.451420
$832$	0	0
$833$	15.0185	0.520359
$834$	0	0
$835$	−0.168236	−0.00582204
$836$	0	0
$837$	−1.21322	−0.0419351
$838$	0	0
$839$	−23.4299	−0.808889	−0.404444	−	0.914563i	$-0.632535\pi$
$839$	−23.4299	−0.808889	−0.404444	+	0.914563i	$0.632535\pi$
$840$	0	0
$841$	−27.3335	−0.942534
$842$	0	0
$843$	14.6624	0.504998
$844$	0	0
$845$	0.213135	0.00733208
$846$	0	0
$847$	−6.53694	−0.224612
$848$	0	0
$849$	4.37195	0.150045
$850$	0	0
$851$	58.8136	2.01610
$852$	0	0
$853$	−30.5289	−1.04529	−0.522645	−	0.852551i	$-0.675055\pi$
$853$	−30.5289	−1.04529	−0.522645	+	0.852551i	$0.675055\pi$
$854$	0	0
$855$	−0.140763	−0.00481399
$856$	0	0
$857$	13.3434	0.455803	0.227902	−	0.973684i	$-0.426814\pi$
$857$	13.3434	0.455803	0.227902	+	0.973684i	$0.426814\pi$
$858$	0	0
$859$	−33.1185	−1.12999	−0.564994	−	0.825095i	$-0.691122\pi$
$859$	−33.1185	−1.12999	−0.564994	+	0.825095i	$0.691122\pi$
$860$	0	0
$861$	1.54592	0.0526848
$862$	0	0
$863$	−38.8602	−1.32282	−0.661409	−	0.750026i	$-0.730041\pi$
$863$	−38.8602	−1.32282	−0.661409	+	0.750026i	$0.730041\pi$
$864$	0	0
$865$	−0.715071	−0.0243131
$866$	0	0
$867$	−11.8119	−0.401153
$868$	0	0
$869$	−6.21954	−0.210983
$870$	0	0
$871$	69.3495	2.34982
$872$	0	0
$873$	11.8338	0.400513
$874$	0	0
$875$	−0.223085	−0.00754166
$876$	0	0
$877$	48.6809	1.64384	0.821919	−	0.569604i	$-0.192904\pi$
$877$	48.6809	1.64384	0.821919	+	0.569604i	$0.192904\pi$
$878$	0	0
$879$	−23.6309	−0.797052
$880$	0	0
$881$	9.27848	0.312600	0.156300	−	0.987710i	$-0.450043\pi$
$881$	9.27848	0.312600	0.156300	+	0.987710i	$0.450043\pi$
$882$	0	0
$883$	20.7522	0.698368	0.349184	−	0.937054i	$-0.386459\pi$
$883$	20.7522	0.698368	0.349184	+	0.937054i	$0.386459\pi$
$884$	0	0
$885$	−0.0557306	−0.00187336
$886$	0	0
$887$	47.6183	1.59887	0.799433	−	0.600756i	$-0.205134\pi$
$887$	47.6183	1.59887	0.799433	+	0.600756i	$0.205134\pi$
$888$	0	0
$889$	−4.75398	−0.159443
$890$	0	0
$891$	−0.863742	−0.0289365
$892$	0	0
$893$	16.0408	0.536783
$894$	0	0
$895$	−0.0636593	−0.00212790
$896$	0	0
$897$	−35.9588	−1.20063
$898$	0	0
$899$	−1.56619	−0.0522355
$900$	0	0
$901$	−17.4762	−0.582218
$902$	0	0
$903$	0.00101097	3.36430e−5 0
$904$	0	0
$905$	−0.207436	−0.00689541
$906$	0	0
$907$	22.5067	0.747323	0.373662	−	0.927565i	$-0.378102\pi$
$907$	22.5067	0.747323	0.373662	+	0.927565i	$0.378102\pi$
$908$	0	0
$909$	−15.7416	−0.522117
$910$	0	0
$911$	23.9254	0.792685	0.396343	−	0.918103i	$-0.370279\pi$
$911$	23.9254	0.792685	0.396343	+	0.918103i	$0.370279\pi$
$912$	0	0
$913$	11.5985	0.383855
$914$	0	0
$915$	−0.122518	−0.00405033
$916$	0	0
$917$	−0.0209969	−0.000693378 0
$918$	0	0
$919$	40.6929	1.34234	0.671168	−	0.741305i	$-0.265793\pi$
$919$	40.6929	1.34234	0.671168	+	0.741305i	$0.265793\pi$
$920$	0	0
$921$	16.8530	0.555325
$922$	0	0
$923$	56.2489	1.85145
$924$	0	0
$925$	−35.7222	−1.17454
$926$	0	0
$927$	−11.4296	−0.375396
$928$	0	0
$929$	−1.00948	−0.0331198	−0.0165599	−	0.999863i	$-0.505271\pi$
$929$	−1.00948	−0.0331198	−0.0165599	+	0.999863i	$0.505271\pi$
$930$	0	0
$931$	26.5197	0.869149
$932$	0	0
$933$	−8.09954	−0.265167
$934$	0	0
$935$	0.0688540	0.00225177
$936$	0	0
$937$	−12.0701	−0.394314	−0.197157	−	0.980372i	$-0.563171\pi$
$937$	−12.0701	−0.394314	−0.197157	+	0.980372i	$0.563171\pi$
$938$	0	0
$939$	−27.5042	−0.897566
$940$	0	0
$941$	−12.1525	−0.396161	−0.198080	−	0.980186i	$-0.563471\pi$
$941$	−12.1525	−0.396161	−0.198080	+	0.980186i	$0.563471\pi$
$942$	0	0
$943$	19.9575	0.649906
$944$	0	0
$945$	0.0223113	0.000725786 0
$946$	0	0
$947$	−19.0889	−0.620306	−0.310153	−	0.950687i	$-0.600380\pi$
$947$	−19.0889	−0.620306	−0.310153	+	0.950687i	$0.600380\pi$
$948$	0	0
$949$	12.4559	0.404335
$950$	0	0
$951$	−22.0581	−0.715284
$952$	0	0
$953$	−20.4334	−0.661904	−0.330952	−	0.943648i	$-0.607370\pi$
$953$	−20.4334	−0.661904	−0.330952	+	0.943648i	$0.607370\pi$
$954$	0	0
$955$	0.0298370	0.000965503 0
$956$	0	0
$957$	−1.11504	−0.0360440
$958$	0	0
$959$	−8.79377	−0.283966
$960$	0	0
$961$	−29.5281	−0.952519
$962$	0	0
$963$	−8.44980	−0.272291
$964$	0	0
$965$	−0.00958893	−0.000308679 0
$966$	0	0
$967$	41.4076	1.33158	0.665790	−	0.746140i	$-0.268095\pi$
$967$	41.4076	1.33158	0.665790	+	0.746140i	$0.268095\pi$
$968$	0	0
$969$	9.16118	0.294300
$970$	0	0
$971$	−15.5077	−0.497667	−0.248834	−	0.968546i	$-0.580047\pi$
$971$	−15.5077	−0.497667	−0.248834	+	0.968546i	$0.580047\pi$
$972$	0	0
$973$	3.96922	0.127247
$974$	0	0
$975$	21.8407	0.699462
$976$	0	0
$977$	43.6653	1.39698	0.698489	−	0.715621i	$-0.253856\pi$
$977$	43.6653	1.39698	0.698489	+	0.715621i	$0.253856\pi$
$978$	0	0
$979$	8.67687	0.277314
$980$	0	0
$981$	7.81853	0.249626
$982$	0	0
$983$	16.1033	0.513615	0.256807	−	0.966463i	$-0.417329\pi$
$983$	16.1033	0.513615	0.256807	+	0.966463i	$0.417329\pi$
$984$	0	0
$985$	−0.177483	−0.00565508
$986$	0	0
$987$	−2.54250	−0.0809287
$988$	0	0
$989$	0.0130514	0.000415011 0
$990$	0	0
$991$	22.2551	0.706957	0.353479	−	0.935443i	$-0.384999\pi$
$991$	22.2551	0.706957	0.353479	+	0.935443i	$0.384999\pi$
$992$	0	0
$993$	22.3009	0.707696
$994$	0	0
$995$	0.150373	0.00476714
$996$	0	0
$997$	−14.7582	−0.467396	−0.233698	−	0.972309i	$-0.575083\pi$
$997$	−14.7582	−0.467396	−0.233698	+	0.972309i	$0.575083\pi$
$998$	0	0
$999$	7.14619	0.226096

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.10	✓	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.0349978 q^{5} +0.637505 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.0349978 q^{5} +0.637505 q^{7} +1.00000 q^{9} -0.863742 q^{11} -4.36921 q^{13} +0.0349978 q^{15} -2.27774 q^{17} -4.02205 q^{19} +0.637505 q^{21} +8.23006 q^{23} -4.99878 q^{25} +1.00000 q^{27} +1.29094 q^{29} -1.21322 q^{31} -0.863742 q^{33} +0.0223113 q^{35} +7.14619 q^{37} -4.36921 q^{39} +2.42495 q^{41} +0.00158582 q^{43} +0.0349978 q^{45} -3.98820 q^{47} -6.59359 q^{49} -2.27774 q^{51} +7.67263 q^{53} -0.0302291 q^{55} -4.02205 q^{57} -1.59240 q^{59} -3.50074 q^{61} +0.637505 q^{63} -0.152913 q^{65} -15.8723 q^{67} +8.23006 q^{69} -12.8739 q^{71} -2.85083 q^{73} -4.99878 q^{75} -0.550640 q^{77} +7.20069 q^{79} +1.00000 q^{81} -13.4282 q^{83} -0.0797159 q^{85} +1.29094 q^{87} -10.0457 q^{89} -2.78539 q^{91} -1.21322 q^{93} -0.140763 q^{95} +11.8338 q^{97} -0.863742 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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